OF COLLISION, AND OF ENERGY. 



51 



the cutvalure must be constant, in order that the tension of 

 the superficial fibrer. may be equal ; and the breadth must 

 be as the distance of the line of application of the force ; 

 that is, as the ordinate of a circular arc, or, when the curva- 

 ture is smkll, it must be equal to the ordinate of another 

 circular arc, of which the chord is equal to the axis. 



341. Theoeem. If a column be cut out 

 of a planT< of equable breadth, and the out- 

 line limiting its depth be composed of two 

 triangles, joined at their bases, the tension of 

 the surfaces produced by a longitudinal 

 force, will be every where equal, when the 

 radius of curvature at the middle becomes 

 equal to half the length of the column ; and 

 in this case the curve will be a cycloid. 



For in the cycloid, the radius of curvature varies as the 

 distance, in the curve, from its origin, or as the square root 



of the ordinate a, and if the depth i be as this distance, a 

 will vary as lb, and the curvature, which is proportional to 



— , will be always as -, and the tensioa wUl be equable 



throughout. In every cycloirf the radius of curvature at 

 the middle point is half of the length. 



Scholium. When the curvature at the mTddle differs 

 from that of the cycloid, the figure of the column becomes of 

 more difficult investigation. It may however be delineated 

 mechanically, making both the depth of the column and 

 its radius of curvature proportional always to ^a. If the 

 breadth of the column vary in the same proportion as the 

 depth, they must both be every where as the cube rootof a. 



SECT. X. OF COLLISION, AND OF ENERGY. 



342. Theorem. When, two elastic bodies 

 approach each other with a uniform motion, 

 until at a certain point a repulsive force com^ 

 mences, their relative velocities, in their re- 

 turn back from that point, will again be uni- 

 form, and equal to what they were, but in a 

 contrary direction, 



Fcr according to the definition of elastic bodies, their, 

 forces are always the same at the same distances from the 

 centres, since they depend on the degree of compression. 

 And if two bodies act reciprocally, so as to change the di- 

 Tcci ion of each other's motions, by any forces which are 

 Jtlways the same at the same distance,'tbeii relative velocities 



in approaching and receding will be equal at equal distances. 

 For since the velocity generated in describing each ele- 

 ment of the distance in returning, is equal to that which 

 was destroyed while the same element of space was describ- 

 ed in approaching, the whole velocities at any equal dis- 

 tances must also be equal. 



Scholium. Bodies which communicate motion without 

 a permanent repulsive force, or in circumstances which 

 more or less prevent its action, are called more or less in- 

 elastic. 



343. Theorem. When two elastic bodies 

 meet each other directly, their velocities after 

 collision are equal to twice the velocity of 

 the common centre of inertia, diminished by 

 their respective velocities. 



For the motion of the centre of inertia remains unaltered, 

 and the motions of the bodies with respect to each other 

 and with respect to the centre of inertia being, after colli- 

 sion, equal and in contrary directions, the velocity of each, 

 must be changed by twice the difference of its velocity and 

 that of the centre of inertia, and will therefore become 

 equal to twice the velocity of the centre of inertia diminish- 

 ed by its own velbcity. 



344. Theorem. When two equal elastic 

 bodies meet each other directlyj their motions 

 will be e.xchangedi 



For twice the velocity of the centre ofinertia is here the. 

 sum of the velocities ; therefore either deducted from this 

 will leave a remainder equal to the other, for the motion of. 

 the body to which it belongs. - 



345. Theorem. An elastic body striking 

 a larger one at rest, is partially reflected, 

 and a body striking a smaller one, continues- 

 to move forwards. 



For the velocity in the first case is greater than twice 

 that of the centre of inertia, in the second smaller. 



346. Theorem. When the impulse of 

 an elastic body is communicated to another 

 through a series of bodies differing infinitely 

 httle from each other in bulk, the momen- 

 tum of the last is to that of the first' in the 

 siibduplicate ratio of their bulks. 



Let the first be 1 — T, the second 1 -f-a-, and the velocity of ; 

 the first 1 ; then the velocity of the centre of gravity will 



1— a: 

 be — - — ,. and the velocity of the. second after the. inif. 



